While filling the bundles in the Community Center is often a player's first priority and can be almost completed within their first year, the museum is a bit trickier than that. Unfortunately, there isn't one quick way to find all of the artifacts that are supposed to fill the building's collection, as many of the pieces aren't found or mined in one particular place.
This means that the best way to fill the museum is to pour your time into the game.
That being said, there are a couple of tricks you can use to make the process a bit easier, hopefully resulting in fewer duplicate artifacts.
First, be sure to keep your hoe on you at all times. As you explore the town, forest, and even the mines, you'll find little spots of worms wiggling in the ground. These artifact spots can be dug up with a hoe, and while there's no guarantee there will always be a new artifact in them, you'll want to dig up every one you find.
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When fishing, you should also try to bring up the treasure chests when you have the opportunity to. It's a separate progress bar from the fishing one, and if you lose the fish you also lose the treasure chest, but if you manage to snag one they often contain artifacts inside.
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You should also gather geodes and artifact troves whenever you can, as those can contain artifacts inside.
Lastly, pay close attention to your luck. If you watch the Fortune Teller every day, she'll tell you what your luck is, increasing your chances of finding something new.